= Gaussian free-energy logarithms at integer singular powers
{c}
{title2=$f_s(t)\sim t^n\log t$}
A Gaussian <determinant> with effective dimension $D$ has a singular free-energy contribution proportional to $t^{D/2}$ after subtraction of cutoff-dependent analytic terms. When $D/2$ is a positive integer, a logarithm replaces the pure power. Differentiating the <determinant> sufficiently often gives a marginal momentum integral proportional to $\log t$. For the anisotropic kernel $q_\parallel^2+q_\perp^4+t$, $D=(d+1)/2$, so logarithms occur at $d=3,7,11,\ldots$. These qualify simple homogeneous-power formulas rather than changing the nominal scaling dimensions.
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